A double mean field equation related to a curvature prescription problem

We study a double mean field–type PDE related to a prescribed curvature problem on compacts surfaces with boundary: {−Δu=2ρ([Formula presented]−[Formula presented]) [Formula presented] in are real parameters, K,h are smooth positive functions on Σ and ∂Σ respectively and ν is the outward unit normal vector to ∂Σ. We provide a general blow–up analysis, then a Moser–Trudinger inequality, which gives energy–minimizing solutions for some range of parameters. Finally, we provide existence of min–max solutions for a wider range of parameters, which is dense in the plane if Σ is not simply connected.